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For every complex number $x$, let $\Vert x\Vert_{\mathbb{Z}}:=\min\{|x-m|:\ m\in\mathbb{Z}\}$. Let $K$ be a number field, let $k\in\mathbb{N}$, and let $\alpha_1,\ldots,\alpha_k$ be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of $\theta\in (0,1)$ such that there are infinitely many tuples $(n,q_1,\ldots,q_k)$ satisfying $\Vert q_1\alpha_1^n+\ldots+q_k\alpha_k^n\Vert_{\mathbb{Z}}

In this paper, we prove an exponential integral formula for the Fourier transform of Bessel functions over complex numbers, along with a radial exponential integral formula. The former will enable us to develop the complex spectral theory of the relative trace formula for the Shimura–Waldspurger correspondence and extend the Waldspurger formula from totally real fields to arbitrary number fields.

We discuss some results and conjectures related to the existence of the non-nilpotent motivic maps $\eta$ and $\mu_9$. To this purpose, we establish a theory of power operations for motivic $H_{\infty}$-spectra. Using this, we show that the naive motivic analogue of the unstable Kahn-Priddy theorem fails. Over the complex numbers, we show that the motivic $T$-spectrum $S[\eta^{-1},\mu_9^{-1}]$ is closely related to higher Witt groups, where $S$ is the motivic sphere spectrum and $\eta$, $\sigma$ and $\mu_9$ are explicit elements in $\pi_{**}(S)$., Comment: Some modifications in section 3. With an appendix on nilpotence in Milnor-Witt K-theory by Marcus Zibrowius

The classification of normal singular cubic surfaces in P 3 \mathbf {P}^3 over a complex number field C \mathbf {C} was given by J. W. Bruce and C. T. C. Wall. In this paper, first we prove their results by a different way, second we provide normal forms of normal singular cubic surfaces according to the type of singularities, and finally we determine automorphism groups on normal singular cubic surfaces with no parameters.

We study the one-to-one analytic maps σ \sigma that send the unit disc into a region G G with the property that λ G ⊂ G \lambda G\subset G for some complex number λ \lambda , 0 > | λ | > 1 0>|\lambda |>1 . These functions arise in iteration theory, giving a model for the self-maps of the unit disk into itself, and in the study of composition operators as their eigenfunctions. We show that for such functions there are geometrical conditions on the image region G G that characterize their rate of growth, i.e. we prove that σ ∈ ⋂ p > ∞ H p \sigma \in \bigcap _{p>\infty }H^p if and only if G G does not contain a twisted sector. Then, we examine the connection with composition operators, and further investigate the no twisted sector condition. Finally, in the Appendix, we give a different proof of a result of J. Shapiro about the essential norm of a composition operator.

If f is a nonzero complex-valued function defined on a finite abelian group A and \hat f is its Fourier transform, then |Supp (f)||Supp {\hat f)| \ge |A|, where Supp (f) and Supp (\hat f) are the supports of f and \hat f. In this paper we generalize this known result in several directions. In particular, we prove an analogous inequality where the abelian group A is replaced by a transitive right G-set, where G is an arbitrary finite group. We obtain stronger inequalities when the G-set is primitive and we determine the primitive groups for which equality holds. We also explore connections between inequalities of this type and a result of Chebotar\"ev on complex roots of unity, and we thereby obtain a new proof of Chebotar\"ev's theorem., Comment: 27 pages

A complex number z z is in the final set of an analytic function f f , as defined by Pólya, if every neighborhood of z z contains zeros of infinitely many f ( n ) {f^{(n)}} . If f f is a Hadamard gap series, then the part of the final set in the open disk of convergence is the origin along with a union of concentric circles.

A Humbert surface is a hypersurface of the moduli space A 2 \mathcal A_2 of principally polarized abelian surfaces defined by an equation of the form a z 1 + b z 2 + c z 3 + d ( z 2 2 − z 1 z 3 ) + e = 0 az_1+bz_2+cz_3+d(z_2^2-z_1z_3)+e=0 with integers a , … , e a,\ldots ,e . We give geometric characterizations of such Humbert surfaces in terms of the presence of certain curves on the associated Kummer plane. Intriguingly this shows that a certain plane configuration of lines and curves already carries all information about principally polarized abelian surfaces admitting a symmetric endomorphism with given discriminant.

In this paper we shall determine all actions of groups of prime order p with p > 5 on Gorenstein del Pezzo (singular) surfaces Y of Picard number 1. We show that every order-p element in Aut(Y) (= Aut(Y), Y being the minimal resolution of Y) is lifted from a projective transformation of P 2 . We also determine when Aut(Y) is finite in terms of K 2 Y , Sing Y and the number of singular members in |-K |. In particular, we show that either |Aut(Y)] = 2 a 3 b for some 1 5, there is at least one element gp of order p in Aut(Y) (hence |Aut(Y)| is infinite).

We give a new characterization of Iitaka's fibration of algebraic varieties associated to line bundles. Introducing an intersection number of line bundles and curves by using the notion of multiplier ideal sheaves, Iitaka's fibration can be regarded as a numerically trivial fibration in terms of this intersection theory.

Let 21 be a finite-dimensional simple Lie algebra over the complex numbers. It is shown that a module is complete (or relatively complete) in the sense of Enright if and only if it is injectively copresented by certain injective modules in the BGG-category O. Let A be the finite-dimensional algebra associated to a block of O. Then the corresponding block of the category of complete modules is equivalent to the category of eAe-modules for a suitable choice of the idempotent e. Using this equivalence, a very easy proof is given for Deodhar's theorem (also proved by Bouaziz) that completion functors satisfy braid relations. The algebra eAe is left properly and standardly stratified. It satisfies a double centralizer property similar to Soergel's combinatorial description of O. Its simple objects, their characters and their multiplicities in projective or standard objects are determined.

If F (x, y) is a binary cubic form with integer coefficients such that F (x, 1) has at least two distinct complex roots, then the equation F (x, y) = 1 possesses at most ten solutions in integers x and y, nine if F has a nontrivial automorphism group. If, further, F (x, y) is reducible over Z[x, y], then this equation has at most 2 solutions, unless F (x, y) is equivalent under GL2(Z)action to either x(x2 − xy − y2) or x(x2 − 2y2). The proofs of these results rely upon the method of Thue-Siegel as refined by Evertse, together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational Diophantine approximation. Along the way, we completely solve all Thue equations F (x, y) = 1 for F cubic and irreducible of positive discriminant DF ≤ 106. As corollaries, we obtain bounds for the number of solutions to more general cubic Thue equations of the form F (x, y) = m and to Mordell’s equation y2 = x3 + k, where m and k are nonzero integers.

Let γ ≡ γ(2n) denote a sequence of complex numbers γ00, γ01, γ10, . . . , γ0,2n, . . . , γ2n,0 (γ00 > 0, γij = γji), and let K denote a closed subset of the complex plane C. The Truncated Complex K-Moment Problem for γ entails determining whether there exists a positive Borel measure μ on C such that γij = ∫ zizj dμ (0 ≤ i + j ≤ 2n) and suppμ ⊆ K. For K ≡ KP a semi-algebraic set determined by a collection of complex polynomials P = {pi (z, z)}mi=1, we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix M (n) (γ) and the localizing matrices Mpi . We prove that there exists a rankM (n)-atomic representing measure for γ(2n) supported in KP if and only if M (n) ≥ 0 and there is some rankpreserving extension M (n+ 1) for which Mpi (n+ ki) ≥ 0, where deg pi = 2ki or 2ki − 1 (1 ≤ i ≤ m).

The known definitions of Farey polytopes and continued fractions are generalized and applied to diophantine approximation in n n -dimensional euclidean spaces. A generalized Remak-Rogers isolation theorem is proved and applied to show that certain Hurwitz constants for discrete groups acting in a hyperbolic space are isolated. The approximation constant for the imaginary quadratic field of discriminant − 15 -15 is found.

We study the distribution of the complex roots of random polynomials of degree n n with i.i.d. coefficients. Using techniques related to Rice’s treatment of the real roots question, we derive, under appropriate moment and regularity conditions, an exact formula for the average density of this distribution, which yields appropriate limit average densities. Further, using a different technique, we prove limit distribution results for coefficients in the domain of attraction of the stable law.

If λ k {\lambda _k} is a sequence of nonzero complex numbers, then we define the zeta regularized product of these numbers, ∏ k λ k \prod \nolimits _k {{\lambda _k}} , to be exp ⁡ ( − Z ′ ( 0 ) ) \exp ( - Z\prime (0)) where Z ( s ) = ∑ k = 0 ∞ λ k − s Z(s) = \sum \nolimits _{k = 0}^\infty {\lambda _k^{ - s}} . We assume that Z ( s ) Z(s) has analytic continuation to a neighborhood of the origin. If λ k {\lambda _k} is the sequence of positive eigenvalues of the Laplacian on a manifold, then the zeta regularized product is known as det ′ Δ \det \prime \Delta , the determinant of the Laplacian, and ∏ k ( λ k − λ ) \prod \nolimits _k {({\lambda _k} - \lambda )} is known as the functional determinant. The purpose of this paper is to discuss properties of the determinant and functional determinant for general sequences of complex numbers. We discuss asymptotic expansions of the functional determinant as λ → − ∞ \lambda \to - \infty and its relationship to the Weierstrass product. We give some applications to the theory of Barnes’ multiple gamma functions and elliptic functions. A new proof is given for Kronecker’s limit formula and the product expansion for Barnes’ double Stirling modular constant.

A self-similar measure on R n {{\mathbf {R}}^n} was defined by Hutchinson to be a probability measure satisfying ( ∗ ) ({\ast }) \[ μ = ∑ j = 1 m a j μ ∘ S j − 1 \mu = \sum \limits _{j = 1}^m {{a_j}\mu \circ S_j^{ - 1}} \] , where S j x = ρ j R j x + b j {S_j}x = {\rho _j}{R_j}x + {b_j} is a contractive similarity ( 0 > ρ j > 1 , R j (0 > {\rho _j} > 1,{R_j} orthogonal) and the weights a j {a_j} satisfy 0 > a j > 1 , ∑ j = 1 m a j = 1 0 > {a_j} > 1,\sum \nolimits _{j = 1}^m {{a_j} = 1} . By analogy, we define a self-similar distribution by the same identity ( ∗ ) ( {\ast } ) but allowing the weights a j {a_j} to be arbitrary complex numbers. We give necessary and sufficient conditions for the existence of a solution to ( ∗ ) ( {\ast } ) among distributions of compact support, and show that the space of such solutions is always finite dimensional. If F F denotes the Fourier transformation of a self-similar distribution of compact support, let \[ H ( R ) = 1 R n − β ∫ | x | ≤ R | F ( x ) | 2 d x , H(R) = \frac {1}{{{R^{n - \beta }}}}\int _{|x| \leq R} {|F(x){|^2}dx,} \] where β \beta is defined by the equation ∑ j = 1 m ρ j − β | a j | 2 = 1 \sum \nolimits _{j = 1}^m {\rho _j^{ - \beta }|{a_j}{|^2} = 1} . If ρ j ν j = ρ \rho _j^{{\nu _j}} = \rho for some fixed ρ \rho and ν j {\nu _j} positive integers we say the { ρ j } \{ {\rho _j}\} are exponentially commensurable. In this case we prove (under some additional hypotheses) that H ( R ) H(R) is asymptotic (in a suitable sense) to a bounded function H ~ ( R ) \tilde H(R) that is bounded away from zero and periodic in the sense that H ~ ( ρ R ) = H ~ ( R ) \tilde H(\rho R) = \tilde H(R) for all R > 0 R > 0 . If the { ρ j } \{ {\rho _j}\} are exponentially incommensurable then lim R → ∞ H ( R ) {\lim _{R \to \infty }}H(R) exists and is nonzero.

This work was motivated by the paper of R. Johnson and J. Moser (see [J-M] in the references) on the one-dimensional almost periodic potentials. Here we study the operator L = − Δ / 2 − q L = - \Delta /2 - q , where q q is an almost periodic function in R d {R^d} . It is shown that some of the results of [J-M] extend to the multidimensional case (our approach includes the one-dimensional case as well). We start with the kernel k ( t , x , y ) k(t,x,y) of the semigroup e − t L {e^{ - tL}} . For fixed t > 0 t > 0 and u ∈ R d u \in {R^d} , it is known (we review the proof) that k ( t , x , x + u ) k(t,x,x + u) is almost periodic in x x with frequency module not bigger than the one of q q . We show that k ( t , x , y ) k(t,x,y) is, also, uniformly continuous on [ a , b ] × R d × R d [a,b] \times {R^d} \times {R^d} . These results imply that, if we set y = x + u y = x + u in the kernel G m ( x , y ; z ) {G^m}(x,y;z) of ( L − z ) − m {(L - z)^{ - m}} it becomes almost periodic in x x (for the case u = 0 u = 0 we must assume that m > d / 2 m > d/2 ), which is a generalization of an old one-dimensional result of Scharf (see [S.G]). After this, we are able to define w m ( z ) = M x [ G m ( x , x ; z ) ] {w_m}(z) = {M_x}[{G^m}(x,x;z)] , and, by integrating this m m times, an analog of the complex rotation number w ( z ) w(z) of [J-M]. We also show that, if e ( x , y ; λ ) e(x,y;\lambda ) is the kernel of the projection operator E λ {E_\lambda } associated to L L , then the mean value α ( λ ) = M x [ e ( x , x ; λ ) ] \alpha (\lambda ) = {M_x}[e(x,x;\lambda )] exists. In one dimension, this (times π \pi ) is the rotation number. In higher dimensions ( d = 1 d = 1 included), we show that d α ( λ ) d\alpha (\lambda ) is the density of states measure of [A-S] and it is related to w m ( z ) {w_m}(z) in a nice way. Finally, we derive a formula for the functional derivative of w m ( z ; q ) {w_m}(z;q) with respect to q q , which extends a result of [J-M].

A linear series ( V , L ) (V,\mathcal {L}) on a curve X X has an N N -fold point along a divisor D D of degree N N if dim ⁡ ( V ∩ H 0 ( X , L ( − D ) ) ) ≥ dim V − 1 \dim (V \cap {H^0}\;(X,\mathcal {L}\,(- D))) \geq \dim \;V - 1 . The dimensions of the families of linear series with an N N -fold point are determined for general curves.

We prove no nontrivial expansion of the field of complex numbers can be obtained from a reduct of the field of real numbers.

The study of quintic surfaces is of special interest because 5 5 is the lowest degree of surfaces of general type. The aim of this paper is to give a classification of the quintic surfaces of general type over the complex number field C {\mathbf {C}} . We show that if S S is an irreducible quintic surface of general type; then it must be normal, and it has only elliptic double or triple points as essential singularities. Then we classify all such surfaces in terms of the classification of the elliptic double and triple points. We give many examples in order to verify the existence of various types of quintic surfaces of general type. We also make a study of the double or triple covering of a quintic surface over P 2 {{\mathbf {P}}^2} obtained by the projection from a triple or double point on the surface. This reduces the classification of the surfaces to the classification of branch loci satisfying certain conditions. Finally we derive some properties of the Hilbert schemes of some types of quintic surfaces.

The application of mathematics to natural phenomena often brings up the question : when a linear vector equation is changed slightly, how much of a change results in the solution? Here "vector" is used to include, for example, vectors in Hubert space and elements of (linear) function spaces; also it is assumedC) that the equation is uniquely solvable. We formulate and answer the question in §2 for a linear equation in a Banach space, then specialize to deduce perturbation limits for linear algebraic systems, infinite systems, and integral equations. Finally we obtain error limitations for certain approximate methods of solving infinite linear systems (method of segments) and integral equations (method of Goursat-Schmidt). It is hoped that these limitations may be useful to applied mathematicians. The methods and results of this paper unify and extend investigations concerning algebraic systems by F. R. Moulton [15](2), Etherington [6] and the author [13, 14]; and on integral equations of Fredholm type and second kind by Tricomi [21]. However, they do not cover perturbation questions associated with characteristic values and characteristic functions, which have been studied by Lord Rayleigh [16, p. 115], Courant [4, p. 296] and Mrs. Adams [l]. §1 is expository, containing as much about abstract vector spaces as is needed for §2. 1. Vector spaces. It will be useful to collect here some facts about normed linear vector spaces ispaces L), Banach spaces ispaces B), and linear transformations. A space L is linear: if vector xEL and a is any complex number, the product ax is defined and axEL; if also y EL, the sum x+y is defined and x+yEL. With each element x of L is associated a non-negative real number ||*||, its norm; ||*|| >0 unless x = @, the zero-element of L; |J@|| =0. The norm has properties of an absolute value: ||a*|| = \a\ -11*11, ||*+y|| a||*||+||y||(We have described a complex space L; in a real space, number a must be real.) A normed linear vector space is a Banach space 5 [2, p. 53] if it is furthermore complete: if {*„} is an infinite sequence of elements of 5, and if II*»—*»||—*0 as m, m—»oo, there exists a vector xo of 5 such that ||*„—*o||—»0 as n—+ oo (strong completeness). Examples of such spaces are listed in Banach's book [2, pp. 10-12, examples 3-10], and several occur in the remainder of this paper.

where c is a scalar matrix. It is well knownf that a relation of this type can not be satisfied by finite matrices. However, in the calculation of commutation formulas for polynomials in p and q no use is made of the fact that c is a scalar but merely that it is commutative with both p and q.% And there do exist pairs of finite matrices x, y of the same order such that xy—yx is not zero and is commutative with both x and y. Such matrices will be called quasi-commutative matrices and either may be said to be quasi-commutative with the other. In a certain sense the algebra of polynomials in a pair of quasi-commutative matrices is homeomorphic to the algebra arising in quantum mechanics. It is hoped to discuss such algebras in some detail in a later paper. In the present paper we shall make a brief study of quasi-commutative matrices whose elements belong to the complex number field. The concept of quasi-commutativity is an extension or generalization of commutativity, and as would be expected, some of the results obtained are generalizations of known theorems concerning commutative matrices. The problem of determining quasi-commutative matrices is that of finding matrices x, y, z (^0) which satisfy the equations

in which the coefficients ap, bp are complex numbers, and z is a complex parameter, have been called J-fractions because of their connection with the infinite matrices known as J-matrices. The theory of J-fractions with real coefficients includes the Stieltjes continued fraction theory and certain of its extensions. In a recent paper, Hellinger and Wall [3 ](1) treated the case where ap is real and bp is an arbitrary complex number with nonnegative imaginary part. In these cases, the J-fraction obviously has the property that all the quadratic forms

Thimm introduced the concept of gap-sheaves for analytic subsheaves of finite direct sums of structure-sheaves on domains of complex number spaces (Definition 9, [13]) and proved that these gap-sheaves are coherent if the subsheaves themselves are coherent (Satz 3, [13]). This concept of gap-sheaves can be readily generalized to analytic subsheaves of arbitrary analytic sheaves on general complex spaces (Definition 1, [12]). All the gap-sheaves of coherent analytic subsheaves of arbitrary coherent analytic sheaves on general complex spaces are coherent (Theorem 3, [12]). The gap-sheaves of a given analytic subsheaf depend not only on the subsheaf itself but also on the analytic sheaf in which the given subsheaf is embedded as a subsheaf. In this paper we introduce a new notion of gap-sheaves which we call absolute gap-sheaves (Definition 3 below). These gap-sheaves arise naturally from the problem of removing singularities of local sections of a coherent analytic sheaf. They depend only on a given analytic sheaf and neither require nor depend upon an embedding of the given sheaf as a subsheaf in another analytic sheaf. We give here a necessary and sufficient condition for the coherence of absolute gap-sheaves of coherent sheaves (Theorem 1 below). This yields some results concerning removing singularities of local sections of coherent sheaves (see Remark following Corollary 2 to Theorem 1). Then we use absolute gap-sheaves to derive a theorem (Theorem 2 below) which generalizes Serre's Theorem on the extension of torsion-free coherent analytic sheaves (Theorem 1, [11]). Finally a result on extensions of global sections of coherent analytic sheaves is derived (Theorem 4 below). Unless specified otherwise, complex spaces are in the sense of Grauert (?1, [5]). If Y is an analytic subsheaf of an analytic sheaf Y on a complex space (X, X), then Y: S denotes the ideal-sheaf f defined by fX ={3 E I lsST c Yx} for x E X. E(9, -) denotes {x E X I x =& }x Supp 3denotes the support of 3C If t E r(X, J), then Supp t denotes the support of t. For x E X, tx denotes the germ of t at x. By the annihilator-ideal-sheaf Q1of Y we mean the ideal-sheaf v defined by /x = {s E Xx I s$x = O} for x E X. If 0: (X, X") -(X', *') is a holomorphic map (i.e. a morphism of ringed spaces) from (X, X") to another complex space (X', k'), then R06(Y) denotes the zeroth direct image of T under O. If fE I(X, X) and x E X, we say thatf vanished at x iffx is not a unit in Xx.

where ao is a complex number and f1(z) is a rational function of which the approximate positions of the zeros and poles are known. As first theorem we give what is a generalization not only of some of our recent resultst for the case that fj(z) = (z ai)-, but also of some of Nagy's results? for the case that the a1 are real and positive. THEOREM 1. Let a1 be complex numbers situated in the same given angular domain of which the vertex is the origin and the aperture is 'y, 0

It is known that if h h is an entire function of exponential type and Z ( h ) = { z k } k = 1 Z(h) = {\{ {z_k}\} _{k = 1}} with | h ′ ( z k ) | ⩾ ε exp ⁡ ( − c | z k | ) |h’({z_k})| \geqslant \varepsilon \exp (- c|{z_k}|) for constants ϵ \epsilon , C C independent of k k , then { z k } k = 1 ∞ \{ {z_k}\} _{k = 1}^\infty is a universal interpolation sequence. That is, given any sequence of complex numbers { a k } k = 1 ∞ \{ {a_k}\} _{k = 1}^\infty such that | a k | ⩽ A exp ⁡ ( B | z k | ) |{a_k}| \leqslant A\,\exp (B|{z_k}|) for constants A , B A,B independent of K K then there exists g g of exponential type such that g ( z k ) = a k g({z_k}) = {a_k} . This note is concerned with finding geometric conditions which make { z k } k = 1 ∞ \{ {z_k}\} _{k = 1}^\infty a universal interpolation sequence for various spaces of entire functions. For the space of entire functions of exponential type a necessary and sufficient condition for { z k } k = 1 ∞ \{ {z_k}\} _{k = 1}^\infty to be a universal interpolation sequence is that ∫ 0 | z k | n ( z k , t ) d t / t ⩽ C | z k | + D , k = 1 , 2 , … \int _0^{|{z_k}|} {n({z_k},t)\,dt/t \leqslant C|{z_k}| + D,k = 1} , 2,\ldots , where n ( z k , t ) n({z_k},t) is the number of points of { z k } k = 1 ∞ \{ {z_k}\} _{k = 1}^\infty in the disc of radius t t about z k {z_k} , excluding z k {z_k} , and C , D C,D are constants independent of k k . Results for the space E ^ ′ = { f entire | | f ( z ) | ⩽ A exp ⁡ [ B | Im ⁡ z | + B log ⁡ ( 1 + | z | 2 ) ] } \hat {\mathcal {E}}^\prime = \{ f\;{\text {entire}}||f(z)| \leqslant A\;\exp [B|\operatorname {Im} z| + B\log (1 + |z|^{2})]\} are given but the theory is not as complete as for the above example.

A generalization of the concept of orthogonal complement is introduced in complete and decomposable complex vector spaces with scalar product. Complementation is a construction in the geometry of Hilbert space which was applied to the invariant subspace theory of contractive transformations in Hilbert space by James Rovnyak and the author [6]. The concept was later formalized by the author [3]. Continuous and contractive transformations in Krein spaces appear in the estimation theory of Riemann mapping functions [4]. It is therefore of interest to know whether a generalization of complementation theory applies in Krein spaces. Such a generalization is now obtained. The results are also of interest in the invariant subspace theory of continuous and contractive transformations in Krein spaces [5]. The vector spaces considered are taken over the complex numbers. A scalar product for a vector space )1 is a complex-valued function (a, b) of a and b in )1 which is linear, symmetric, and nondegenerate. Linearity means that the identity (aa + 3b, c) = a(a, c) + 3(b, c) holds for all elements a, b, and c of )1 when a and , are complex numbers. Symmetry means that the identity (b, a) = (a, b) holds for all elements a and b of )1. Nondegeneracy means that an element a of )1 is zero if the scalar product (a, b) is zero for every element b of )1. Every element b of )1 determines a linear functional bon )1 which is defined by b-a = (a, b) for every element a of )1. The weak topology of )1 is the weakest topology with respect to which bis a continuous linear functional on )1 for every element b of )1. The weak topology of )1 is a locally convex topology having the property that every continuous linear functional on )1 is of the form bfor an element b of )1. The element b is then unique. The antispace of a vector space with scalar product is the same vector space considered with the negative of the given scalar product. A fundamental example of a vector space with scalar product is a Hilbert space. A Krein space is a vector space with scalar product which is the orthogonal sum of a Hilbert space and the antispace of a Hilbert space. Received by the editors October 10, 1986 and, in revised form, February 23, 1987. The results of the paper were presented to the Department of Mathematics, Indiana and Purdue University in Indianapolis, on March 27, 1987, as the Ernest J. Johnson Colloquium. 1980 Mathematics Subject Classification (1985 Revision). Primary 46D05. Research supported by the National Science Foundation. ?1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

A linear series ( V , L ) (V,\,\mathcal {L}) on a curve X X has an n n -fold point along a divisor D D of degree n n if dim ⁡ ( V ∩ H 0 ( X , L ( − D ) ) ) ⩾ dim ⁡ ( V ) − 1 \dim (V \cap {H^0}(X,\,\mathcal {L}( - D))) \geqslant \dim (V) - 1 . The linear series has a cusp of order e e at a point P P if dim ⁡ ( V ∩ H 0 ( X , L ( − ( e + 1 ) P ) ) ) ⩾ dim ⁡ ( V ) − 1 \dim (V \cap {H^0}(X,\,\mathcal {L}( - (e + 1)P))) \geqslant \dim (V) - 1 . Linear series with cusps and n n -fold points are shown to exist if certain inequalities are satisfied. The dimensions of the families of linear series with cusps are determined for general curves.

We study the completeness in L 2 ( R ) {L_2}(R) of sequences of the form { f ( c n − t ) } \{ f({c_n} - t)\} , where { c n } \{ {c_n}\} is a sequence of distinct real numbers. A Müntztype theorem is proved, valid for a large class of functions and, in particular, for f ( t ) = exp ⁡ ( − t 2 ) f(t) = \exp ( - {t^2}) .

1. The usual theory relates to systems of numbers =,a,e, in which the co6rdinates ai range independently over all real nulmbers or else over all ordinary complex numbers; for example, the real quaternion system, or the complex quaternion system. As an obvious generalization,t the co6rdinates may range independently over all the marks of any field F; for example, the rational quaternion system. As a further generalization, the sets of co6rdinates al, *, a in the various numbers of a system may include only a part of the sets bl, -, b , each b ranging independently over F; for example, the integral quaternion system. The various coordinates a1, , a. need not have the same range; for example, the numbers (a + 2b V/2) el + (c + 4d 12 ) e2 (a, b, c, d arbitrary integers)

quantum field theory. Although few nonspecialists have had opportunity to become familiar with the language of modern pure mathematics, quantum theory seems to have reached a point where it must use that language if it is to find a genuine escape from the divergence difficulties. Divergence can not be properly coped with when convergence itself has never been rigorously defined. In the classical analysis of real and complex numbers, results, even correct results, can be obtained by algebraic manipulation of formal power series; but these numbers are not just algebras, they are topological algebras, and only with Cauchy's introduction of the epsilon-delta treatment was mathematics provided an explicit method of separating sense from nonsense. Similarly, in the modern analysis of infinite-dimensional algebras results can be obtained by algebraic manipulation of formal expressions, but these results often require topological justification. One standard way of introducing a topology into the algebra of observables is to make them operators on a Hilbert space. This method, which does not seem to be extensively employed in quantum electrodynamics, can be used to construct a mathematically rigorous formalism the manipulation of which is directly followable by one's physical intuition. This construction requires the exercise of two dissimilar disciplines, mathematics and physics, so the exposition is divided into two parts upon which relative emphasis can be adjusted to suit individual tastes. In particular, physicists can greatly simplify the mathematics by ignoring: (1) operator-domain considerations (as is done here in the derivation of the Yukawa-potential); (2) discussions involving the group algebra of the symmetric group (since only the FermiDirac and Bose-Einstein cases have ever actually occurred); (3) material depending on the simply-connected covering group of the Lorentz group (since it is not needed to derive Maxwell's equations). However, Part I is empty, unmotivated mathematics without Part II; and Part II does not exist without Part I. The two are designed to be read, not consecutively, but in parallel. Sections are numbered accordingly. I would like to thank Professor I. E. Segal for liberal use of his time and advice in the preparation of this paper. It is to be submitted to the Depart

1. The Geometry of Numbers was founded by Minkowski in order to attack certain arithmetical problems, and is normally concerned with lattices over the rational integers. Minkowski himself, however, also treated a special problem over complex quadratic number fields [5], and a number of writers have since followed him. They were largely concerned with those fields which have class-number h= 1; and this simplification removes many of the characteristic features of the more general case. Hermann Weyl [10] gave a thorough account of the extension of Minkowski's theory of the reduction of quadratic forms to "gauge functions" over general algebraic number fields and quaternion algebras, and we shall follow part of his developments, though our definition of a lattice is quite different. The desirability of extending the Geometry of Numbers to general algebraic number fields was emphasized by Mahler in a seminar at Princeton. In this paper we shall carry out this program, extending the fundamental results of Mahler [4] to our more general case and applying them to specific problems. Certain new ideas are necessary, but much of this paper must be regarded as expository. In particular, when the proof of a result is essentially analogous to that for the real case we have merely given a reference to the latter. 2. Let K be an algebraic extension of the rationals of degree m. We regard K as an algebra over the rationals, which we can extend to an algebra K* over the reals. It is well known that K* is commutative and semi-simple (being in fact isomorphic to the direct sum of r copies of the reals and s copies of the complex numbers, where r and 2s are the number of real and complex conjugates of K); and the integers of K* are just those of K. We now define the n-dimensional space Kn over K as being the set of ordered n-tuples of elements in K*. Any tCK* is of the form t=x1i1+ * * * +xwmU, where the x, are real and co,, * * *, com is an integral basis for K; and hence there is a natural map of Kn onto Rmn in which each component t is mapped onto m of the components of the point in Rmn, namely xi, x, m as above. We can define a metric and a measure in Kn by means of those in Rmn, with the above map, and so Kn is a locally compact complete metric space.

The recogllition of the abstract identity of geonletry and analysis, which results from the llotion of coordinates on the one hand and from the classical work of VON STAUDT t on the algebra of throws on the other, and which recent work oll the foundations of geometry has ftllly established, has brought with it a broader conception of the colltellt of geometry. It has meant llot only the introduction of imaginary elements alld the resulting conception of a complex space (of any number of dimensions), but it has also led to the consideration of geometries with respect to any number-system (finite or infinite), i. e., of spaces the elemellts of which may be determined by sets of numbers (coordinates) belonging to a givell number-system. t An important result of the recognition of the identity referred to is the etnphasis it places on the possibility of using geometric or synthetic methods in the solution of analytic problems. It seems likely that the fact that such methods have received comparatively little attention hitherto has resulted in a loss of power. The present paper, it is hoped, will telld to substantiate this assertion. We are llere concerned with certaill fundamental problems in the projective geotnetry on a comptex plane, i. e., a plane the points of which are determined by sets of llomogeneous coordinates (xl, X2, X3), where the xi are any ordinary complex numbers. not all zero. Though the illvestigation is essentially geometric alld the results are susceptible of immediate application to problems of importance in geoInetry, these results are of even greater interest in the theory of functions of two complex variables. To make this clear we shall glance briefly at the corresponding problems on a complex line, tlle results of which are well-known ill the theory of functions

1. A summability matrix is called conservative if it attaches a limit to, that is, sums, every convergent sequence. If moreover this limit is a fixed multiple m of the ordinary limit of the sequence the matrix is called multiplicative m. If a matrix A is multiplicative m then the matrix kA, where k $0 is a number, is multiplicative km and sums exactly the same sequences as A. Thus it is immaterial to specify m except to say whether or not it is zero. The dichotomy of matrices into those for which m =0, m S 0 is well known to be significant. A single example is the theorem of Steinhaus [1 ] (1) that if m #0 a multiplicative m matrix cannot sum all bounded sequences, the result being false if m = 0. The principal object of this paper is to extend this classification into the whole set of conservative matrices. This will be separated into the subclasses of co-regular and co-null matrices; the division of multiplicative matrices induced will be that mentioned above. It will then be shown that a class of theorems which have been proved about multiplicative matrices can be so generalized as to apply to conservative matrices in general. The value of the classification will appear in that certain results in which the condition m 0 plays an essential role will hold for co-regular but not for co-null matrices. Some results are new even for multiplicative matrices, for example, the specialization of Theorem 2.0.3. 1.1. Preliminary. Let A = (ac,k) be a matrix of complex numbers and x = {xn4 a complex sequence, then y =Ax is called the transformed sequence where in the multiplication x and y are treated as column vectors, thus Y= {yn} where yn= Z_o ankXk=An(x). Then, if it exists, lim yn=lim An(x) is called A(x). Finally the domain of the functional A(x), that is, the set of sequences x such that Ax is convergent, is called the summability field of A and written (A). We shall denote the identity matrix by I so that (I) is the set of convergent sequences. Setting, after Brudno (1), jAj =supn Ek= ank and denoting by F the set of sequences i= {I, 1, 1, . . . }, Io= {1, 0, 0, . . . }, 8'= {0, 1, 0, . . *} . . , Sk, * * * , we have the classical result: the matrix A is conservative if and only if (i) |jA|| < Xo and (ii) FC(A). Denoting A(Sk) = limn,O0 ank by ak we can easily show that

This paper is concerned with the theory of ideals in the algebra L1 of integrable functions on a locally compact abelian group. After some preliminaries an analytical proof is given of the known theorem that an analytic function of a Fourier transform represents again a Fourier transform (p. 406). Then, in part I, the continuous homomorphisms of closed ideals I of L1 upon C, the field of complex numbers, are studied. Any such homomorphism is given by a Fourier transform and, if I o is its kernel, the quotient-algebra I/Io, normed in the usual way, is not only algebraically isomorphic, but also isometric with C (Theorem 1.2). Another result states that homomorphic groups have homomorphic L'-algebras and that a corresponding property of isometry holds (Theorem 1.3). In part II, which may be read independently of part I, a theorem of S. Mandelbrojt and S. Agmon, which generalizes Wiener's theorem on the translates of a function in Ll, is extended to groups (Theorem 2.2). Several generalizations of Wiener's classical theorem have been published in the past few years; references to the literature are given on p. 422. The rest of part II is devoted to some applications (pp. 422-425). In conclusion it should be said that the work is carried out in abstract generality, with the methods, and in the spirit, of analysis, which is then applied to algebra. To Professors S. Mandelbrojt, B. L. van der Waerden, and A. Weil I owe my mathematical education. The inspiration which I have received in their lectures, in letters, and above all in personal contact, is at the base of this work ; may I here express my gratitude.

Introduction. The theories of hypercomplex numbers and of continuous groups were first explicitly connected by POINCAR4t in 1884 with the statement that the problem of complex numbers was reduced to that of finding all the linear continuous group of substitutions in n variables of which the coefficients are linear functions of n arbitrary parameters, and since that time the advance in the theory of hypercomplex numbers has been largely suggested by the theory of continuous groups. In 1889 STUDYf and SCHEFFERS? developed the relation between these theories to a considerable extent, and the latter in 1891 arrived at a complete enumeration of systems in less than six units which are inequivalent (of different "' Typus "), non-reciprocal, irreducible, and which possess moduli. Previously (1889) STUDY?f had enumerated all inequivalent systems with moddli in less than five units without direct use of the theory of continuous groups, and in 1890 ROHR** continued the work through systems in five units. The problem of enumerating hypercomplex number systems had been attacked by BENJAMIN PEIRCE about twenty years before the investigations of STUDY and SCHEFFERS. His results were not printed, however, until after his death.tt With the methods of the European investigators in mind, I have else

where { xi } is a sequence of complex elements and the Kn,i are complex numbers, has been widely studied, and the conditions which must be fulfilled by the Kn,i in order that the property of convergence of the sequence may remain invariant were given by Schur [I ].t In recent studies by Hurwitz [2, 3] and Knopp [4] modes of measuring the divergence of bounded sequences were given, and the conditions on the Kn,i were found under which the divergence of the sequence { yn } is no greater than that of {Xn}. In this paper the effects of the transformations will be investigated with fewer restrictions on the Kn,i than those imposed by earlier writers. The problem will be approached by means of the new concept of the limit circle defined as follows: The limit circle of a bounded sequence of complex elements is the (unique) circle of least radius which contains within or on its boundary the limit points of the sequence. The limit circle of a bounded function F(y) of the complex variable y as y->t (finite or infinite) is analogously defined in terms of the limit points of F(y) as y->t; this concept will be used in the study of transformations of sequences and functions into functions. 2. Sequence to function transformations. Instead of the transformation mentioned in the introduction we shall study the following more general transformation S. Let T be a set of points in the complex plane having a limit point to (finite or infinite) not belonging to T. We shall speak of a point t in T as being sufficiently advanced if for some a >0, jt toI < when to is finite, or j i/tI < 5 when to is infinite. Then let Ki(t) be a set of complex numbers defined for i= 1, 2, ... , and each t in T, and such that

Let F \mathfrak {F} be a compact subset of the family A \mathcal {A} of functions analytic in Δ = { z : | z | > 1 } \Delta = \{ z:\;|z| > 1\} , and let L \mathcal {L} be a continuous linear operator of order zero on A \mathcal {A} . We show that if the extreme points of the closed convex hull of F \mathcal {F} is the set { f 0 ( x z ) } ( | x | = 1 ) \{ {f_0}(xz)\} (|x| = 1) , then L ( f ) \mathcal {L}(f) is hull subordinate to L ( f 0 ) \mathcal {L}({f_0}) in Δ \Delta . This generalizes results of R. M. Robinson corresponding to families F \mathcal {F} of functions that are subordinate to ( 1 + z ) / ( 1 − z ) (1 + z)/(1 - z) or to 1 / ( 1 − z ) 2 1/{(1 - z)^2} . Families F \mathcal {F} to which this theorem applies are discussed and we identify each such operator L \mathcal {L} with a suitable sequence of complex numbers. Suppose that Φ \Phi is a nonconstant entire function and that 0 > | z 0 | > 1 0 > |{z_0}| > 1 . We show that the maximum of Re ⁡ { Φ [ log ⁡ ( f ( z 0 ) / z 0 ) ] } \operatorname {Re} \{ \Phi [\log (f({z_0})/{z_0})]\} over the class of starlike functions of order a is attained only by the functions f ( z ) = z / ( 1 − x z ) 2 − 2 α , | x | = 1 f(z) = z/{(1 - xz)^{2 - 2\alpha }},\;|x| = 1 . A similar result is obtained for spirallike mappings. Both results generalize a theorem of G. M. Golusin corresponding to the family of starlike mappings.

Our aim is, first, to present two realizations of the space 1* of all bounded complex linear functionals on 21, and, second, to use these realizations for the study of positive definite functions which are not necessarily continuous. In this fashion, we obtain a generalization of Bochner's representation theorem for continuous positive definite functions as well as various facts concerning positive definite functions and their structure. 0.2 Throughout the present paper, the symbol R denotes the real numbers, considered either as an additive group or as a field; K the field of complex numbers; T the multiplicative group of complex numbers of absolute value 1; Tm the complete Cartesian product of m groups each identical with T, m being any cardinal number greater than 1. The characteristic function of a subset B of a set X is denoted by XB. If G is any locally compact Abelian group, we denote the group of all continuous characters of G by the symbol G]. G] is given the usual compact-open topology. If X is any topological space, we denote the set of all complex-valued continuous functions on X which are bounded in absolute value by the symbol C(X). The space of all trigonometric polynomials 0.1.1 is denoted by 93; the space of all almost periodic continuous functions on R by 2f. For a normed complex linear space V, we denote the space of all bounded complex linear functionals on V by the symbol V*.

In this Memorandum the author presents a technique for the solution of certain integral equations, on a finite interval of the real line, that arise in the theories of neutron transport, radiative transfer, gas dynamics, and electromagnetic wave refraction. The kernels in the equations studied are functions of the difference of the arguments and involve an exponential factor. Existence, uniqueness, and computational aspects are considered. (Author)

Two pairs of n-ary Hermitian forms with nXn matrices A, B and C, D with elements in the complex field are equivalent if there exists a nonsingular matrix T such that T'A T = C and T'BT = D, where T' is the conjugate-transpose of T. As is usual in the study of equivalence of pairs of matrices the work divides itself into the consideration of the non-singular and singular cases. These two cases are taken up in Parts II and I respectively. In the non-singular case the rank of pA +a-B is n except for special values of p and o-. It has frequently been pointed out that in this case no generality is lost by assuming B is of rank n. In the singular case the rank r of pA +?-B is less than n for all values of p and a-, but as above no generality is lost in assuming that the rank r of B is the maximum rank of pA +?-B. By the elementary divisors of a pair of matrices A, B is meant the elementary divisors of A -NB when B is non-singular, and the elementary divisors of pA +a-B when B is singular but the determinant IpA+a-B is not identically zero in p and o-. In the non-singular case, the well known necessary and sufficient condition for the equivalence in any field of pairs of bilinear forms, or of their corresponding matrices, and for the equivalence in the field of complex numbers of pairs of symmetric matrices is that the pairs have the same elementary divisors. This condition is known to be not sufficient

The term linear associative algebra, introduced by BENJAMIN PEIRCE, has the same significance as the term system of (higher) complex numbers. t In the usual theory of complex numbers, the co6rdinates are either real numbers or else ordinary complex quantities. To avoid the resulting double phraseology and to attain an evident generalization of the theory, I shall here consider systems of complex numbers whose co6rdinates belong to an arbitraryfield F. I first give the usual definition by means of a multiplication table for the n units of the system. It employs three postulates, shown to be independent, relating to n3 elements of the field F. The second definition is of abstract character. It employs four independent postulates which completely define a system of complex numbers. The first definition may also be presented in the abstract form used for the second, namely, without the explicit use of units. The second definition may also be presented by means of units. Even aside from the difference in the form of their presentation, the two definitions are essentially different.

We give a new proof in positive characteristic of Castelnuovo's criterion of rationality of algebraic surfaces. We use crystalline cohomology and the de Rham-Witt complex as a substitute for the transcendeal methods of Kodaira. Let X be a nonsingular complete algebraic surface over an algebraically closed field k. Let pa(X) = X(Ox) 1 and let Pn(X) = ho(X, X where Kx is the canonical line bundle. These are birational invariants of X. THEOREM 1. X is rational (birationally isomorphic to the projective plane) if and only if Pa(X) = P2(X) = 0. Over the complex numbers, this is a classical result of Castelnuovo. A modem proof was given by Kodaira [10]. Zariski [16] gave the first proof in characteristic p > 0; other proofs were given by M. Artin (unpublished) and G. Kurke [11]. In this paper, we give a short proof of the theorem in positive characteristic. Our proof is closely related to Artin's; however, by using crystalline cohomology and the de Rham-Witt complex, we are able to give a better approximation in characteristicp to the elegant Kodaira proof in characteristic zero. Our debt to the work of Nygaard [13] and Artin and Swinnerton-Dyer [3] will be obvious. After the first version of this paper was written, I learned that S. Mori has also given a new proof of Castelnuovo's criterion, using his theory of extremal rational curves. I believe that this proof complements his, and have indicated briefly in certain places how (according to taste) one may substitute arguments from his proof for those given here. PROOF OF THEOREM 1. Serre [14] has given an exposition of Kodaira's proof which shows (in all characteristics) that if X is a nonsingular complete algebraic surface free from exceptional curves of the first kind withpa(X) = P2(X) = 0, then either X is rational, or X satisfies (1) Pic(X) is an infinite cyclic group generated by Kx, (2) I-KI consists of irreducible curves of arithmetic genus 1, and diml-KI > 1. Received by the editors December 14, 1979 and, in revised form, February 19, 1980. AMS (MOS) subject classifications (1970). Primary 14J10; Secondary 14F30.

Small ("spatial") free products of C*-algebras are constructed. Under certain conditions they have properties similar to those proved by Paschke and Salinas for the algebras C,*(GI * G2) where G1, G2 are discrete groups. The freeproduct analogs of noncommutative Bernoulli shifts are discussed. 0. Introduction. Let K be a field. Consider the category of unital algebras over K. It is well known that this category admits coproducts: free products of algebras [2]. Heuristically, the free product of algebras is the algebra generated by them, with no relations except for the identification of unit elements. If K = C, the complex numbers, and we consider unital * -algebras, we can easily define a * -operation on the free products. Let A, B be unital C*-algebras, and A * B their free product, which is a unital *-algebra. The question arises: in what ways may one define a pre-C* norm on A * B that extends the norms on A and B? Guided by analogy with tensor products, we expect to have a choice among many pre-C* norms, giving rise to many "C* free products" of A and B. One natural norm is 1c I I = sup{ 1IT(c)IH: ST * -representation of A * B). The * -representations of A * B are in 1-1 correspondence with pairs of * representations of A and B, which act on the same Hilbert space. Let A * B be the completion of A * B in this norm. It is easy to see that this construction defines a coproduct in the category of C*-algebras, and that A * B is the "biggest free product" of A and B, analogous to the biggest tensor product A 0 B. If G 1, G2 are discrete groups we obtain C*(G1) * C*(G2) C*(GI * G2) where G1 * G2 is the free product group, and this is analogous to the relation C*(G1) 0 C*(G2) C*(G1 X G2). This paper is motivated by the question: Is there a "smallest C*-product", A*B, in analogy to the smallest tensor product, A * B, satisfying a relation Cr*(GI * G2) Cr*(GI) * Cr*(G2) Received by the editors January 28, 1981 and, in revised form, May 19, 1981. 1980 Mathematics Subject Classification. Primary 46L05; Secondary 46L55. (1 982 American Mathematical Society 0002-9947/82/00001022/$04.25

Some of the recent work on invariance questions can be re- garded as follows: Characterize those linear operators on Hom(V, V) which preserve the character of a given representation of the full linear group. In this paper, for certain rational characters, necessary and sufficient conditions are described that ensure that the set of all such operators forms a group S. The structure of 2 is also determined. The proofs depend on recent results concerning derivations on symmetry classes of tensors. 1. Statements. Let G be any subgroup of the full linear group GL (n, C) over the complex numbers, and let U denote the linear closure of G in the total

Introduction. Sequences of complex numbers which generate positive semidefinite Toeplitz or Hankel matrices occur naturally in many areas of mathematics and are usually called positive definite sequences. It is well known that a sequence formed from a positive integer power of the terms of a positive definite sequence is always itself a positive definite sequence, but we shall be interested in the sequences of the title, which have the special property that any sequence formed from a positive fractional power of their terms is again a positive definite sequence. These sequences will be characterized and representation formulae will be derived which relate them to certain moment sequences. Their relationship with classical conformal mapping problems will be discussed and the natural interpolation problems for them will be solved. We shall use without further comment the methods and the notation of [5]. As a natural extension of that notation, if A _(aij)!% = 0 is an infinite matrix we shall write A ?O (positive semidefinite) or A >-O on L if and only if every finite section AN=(aIj)1j=O, N=O, 1, 2,..., satisfies AN O or AN?O on LN+1{x E CN+1 N =+1 x= 0}, respectively. 1. Toeplitz forms. If {an}r00 c C is a sequence of complex numbers, we shall